Centrality measures aim to indicate who is important in a network. Various notions of ‘being important’ give rise to different centrality measures. In this paper, we study how important the central vertices are for the connectivity structure of the network, by investigating how the removal of the most central vertices affects the number of connected components and the size of the giant component. We use local convergence techniques to identify the limiting number of connected components for locally converging graphs and centrality measures that depend on the vertex’s neighbourhood. For the size of the giant, we prove a general upper bound. For the matching lower bound, we specialise to the case of degree centrality on one of the most popular models in network science, the configuration model, for which we show that removal of the highest-degree vertices destroys the giant most.

中文翻译：

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基于中心性的顶点去除后随机图的连通性

中心性度量旨在表明谁在网络中很重要。不同的“重要”概念会产生不同的中心性衡量标准。在本文中，我们研究了中心顶点对于连接结构通过研究最中心顶点的移除如何影响连接组件的数量和巨型组件的大小来研究网络的性能。我们用局部收敛技术确定局部收敛图的连接组件的限制数量和取决于顶点邻域的中心性度量。对于巨人的大小，我们证明了一个一般的上限。对于匹配下界，我们专门研究以下情况程度中心性在网络科学中最流行的模型之一上，配置模型，为此我们证明移除最高度的顶点对巨人的破坏最大。